Theorem breq12 4135

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 4133	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 4134	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 466	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   class class class wbr 4130

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131

This theorem is used by:  breq12i  4139  breq12d  4143  breqan12d  4146  posng  4847  isopolem  6028  poxp  6468  rbropapd  6513  ecopover  6907  ecopoverg  6910  ltdcnq  7764  recexpr  8005  ltresr  8206  reapval  8904  ltxr  10177  xrltnr  10181  xrltnsym  10195  xrlttr  10197  xrltso  10198  xrlttri3  10199  xposdif  10284  f1olecpbl  13634  wlk2f  16592  dichmul0orlem3  16755  exmidsbthrlem  17067